I am new to Lightroom and started using it when Lightroom4 beta was released. I was using Lightroom4 beta on a test machine. I have few raw files (NEF) imported and edited in Lightroom. The trial period is expired and Lightroom will not even start. I am planning to install stable version of Lightroom4 on a different machine, and want to move the raw files with all edits to this different machine.

This answers how to do this from inside Lightroom. Is it possible to copy bunch of files from old machine to new machine and Lightroom recognizes it?

You should be able to copy and open the whatever.lrcat file on the new machine (See the File > Open menu item). If you also copy or move the .NEF files, LR will also be able to find those (though you may need to tell it where the images have 'moved' to).